Interaction-driven transport of dark excitons in 2D semiconductors with phonon-mediated optical readout

The growing field of quantum information technology requires propagation of information over long distances with efficient readout mechanisms. Excitonic quantum fluids have emerged as a powerful platform for this task due to their straightforward electro-optical conversion. In two-dimensional transition metal dichalcogenides, the coupling between spin and valley provides exciting opportunities for harnessing, manipulating, and storing bits of information. However, the large inhomogeneity of single layers cannot be overcome by the properties of bright excitons, hindering spin-valley transport. Nonetheless, the rich band structure supports dark excitonic states with strong binding energy and longer lifetime, ideally suited for long-range transport. Here we show that dark excitons can diffuse over several micrometers and prove that this repulsion-driven propagation is robust across non-uniform samples. The long-range propagation of dark states with an optical readout mediated by chiral phonons provides a new concept of excitonic devices for applications in both classical and quantum information technology.

Since the discovery of excitonic resonances in MoS 2 1,2 , transition metal dichalcogenides (TMDs) have emerged as a promising platform for applications in classic and quantum technology. The atomic-size thickness of TMDs imbues excitons with a unique combination of properties that make them attractive in many scientific fields, including optoelectronics, sensing, quantum information, and spintronics 3 . The large binding energy makes excitons in TMDs robust at room temperature, and the combination of their short lifetime and in-plane dipole makes them extremely bright. The broken inversion symmetry and large spin-orbit coupling gives rise to a unique spin-valley coupling that provides an extra degree of freedom for manipulating and storing information 4 . These aspects have been mostly explored in bright excitons, which are bound states formed by an electron and a hole in electron bands with the same spin orientation at the K (or K') valley. On one hand, bright excitons are optically active, with a large oscillation strength and high quantum yield. However, their short lifetime, weak interaction, and short transport range are severe limitations for the potential of TMDs in spintronics and quantum technology 5 . On the other hand, the band structure of TMDs gives rise to a variety of excitonic complexes 6 , and it allows for the formation of several lower-energy dark excitons that cannot directly recombine optically due to the non-conservation of spin or momentum. Spinforbidden excitons are made of carriers in electron bands with opposite spin orientations in the same valley (K and K'), and contrary to their bright counterpart, they possess an out-of-plane transition dipole and two orders of magnitude longer lifetime [7][8][9][10][11] . Despite their dark nature, spin-forbidden excitons can inefficiently recombine optically due to the small spin crossover of the conduction band or through the Rashba Check for updates 1 Photonics Initiative, Advanced Science Research Center, City University of New York, New York, NY 10031, USA. 2 International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 3 Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 4 Department of Electrical Engineering, City College of the City University of New York, New York, NY 10031, USA. 5 Physics Program, Graduate Center, City University of New York, New York, NY 10016, USA. e-mail: ggrosso@gc.cuny.edu effect 12 . Moreover, the spin-flip process required for dark exciton recombination can be controlled by an external magnetic field 13,14 .
Although the dynamics and transport of interlayer excitons in van der Waals heterostructures [15][16][17] and intervalley excitons 18 have been recently investigated, the dynamics and the transport properties of spin-forbidden dark excitons have not been fully explored yet 19 , due to the challenges associated with their detection using far-field spectroscopy techniques 11,20,21 . However, efficient phonon interaction processes can open recombination paths for dark excitons with out-ofplane emission that allow for efficient optical readout without the need for auxiliary external fields.
In this work, we study exciton dynamics by employing highresolution, spatially-resolved photoluminescence (PL) spectroscopy techniques that enable the observation of their transport up to several micrometers. We demonstrate the long-range transport of dark excitons by resolving spectrally their emission away from the excitation position. We show that dark exciton transport is driven by mutual repulsive interaction, and it depends on the exciton density at the excitation region (see Fig. 1a). The information of the quantum state of dark excitons is then read out via the emission of lower-energy excitons resulting from exciton-phonon scattering. Our conclusions are corroborated by two main observations: (i) the size of the dark exciton cloud increases as a function of density; and (ii) dark excitons are shown to diffuse in different energy landscapes, including flat, downhill, and uphill. This last observation can be explained by the strong repulsive interaction energy that overcomes the energy difference between the excitation and detection point. The larger interaction energy of dark excitons compared to the bright counterpart is explained by their higher density. The cartoon in Fig. 1a illustrates the main mechanisms and findings of our experiments. We note that this robust transport is peculiar to dark excitons as, differently from interlayer excitons in heterostructures, the stronger binding energy combined with high interaction energy can compensate for largely inhomogeneous energy landscapes. Moreover, this effect is clearly observable in naturally n-doped samples, indicating that it can be used in simple device geometries without the need for external gating or complex heterostructures in which the exciton properties strongly depend on the twist angle.

PL spectroscopy of encapsulated WS 2 monolayer
Figure 1b (bottom panel) shows the measured emission spectrum at T = 7 K of the WS 2 monolayer encapsulated within two thin hBN layers. In order to study the spin-valley coherence properties and identify the several excitonic complexes in the sample 22,23 , we excite it with a righthanded circularly polarized (σ þ ) laser quasi-resonant with the B exciton and detect the emission of right-handed circularly polarized (σ þ ) or left-handed circularly polarized (σ À ) light. The top panel shows the chirality, defined as the degree of polarization and calculated as ρ = I + ÀI À I + + I À , where I + and I À is the emission intensity of σ þ and σ À polarized light, respectively. The sample shows several excitonic peaks whose energy and polarization degrees are in good agreement with previous studies in n-doped WSe 2 and WS 2 [22][23][24] . Peaks labeled with X 0 , XX 0 , X À T , X À S , and XX À are the bright A neutral exciton, neutral biexciton, triplet trion, singlet trion, and negatively charged biexciton. These exciton complexes are the results of the Coulomb interaction between X 0 and carriers in the K or K' valley. The energy detuning with respect to the bright exciton of all exciton complexes measured in this work is in excellent agreement with previous works and is reported in Supplementary Tables S1 and S3. We identify two peaks in the lowenergy shoulders of the bright trions that have been previously attributed to intervalley momentum-forbidden (I 0 ) and intravalley spin-forbidden (D 0 ) excitons 23,25 . The energy difference between I 0 and D 0 is the result of short-range Coulomb exchange interactions in WS 2 , measured to be ∼ 6.5 meV in agreement with theoretical predictions 26 . The presence of dark excitons is more evident in the chirality plot (top panel of Fig. 1b), where clear features emerge at the low energy side of X À T and X À S . We note that the chirality of D 0 should be suppressed due to its out-of-plane optical transition 23 . The nonvanishing chirality of D 0 that emerges in our measurements is due to the nearby presence of the strong emission of X À S and of a strongly-polarized unknown peak at ∼ 3 meV below. A more careful analysis returns a much lower degree of polarization for D 0 of 7% (see Section 3 of the SI). The peaks labeled with T I and D − have been attributed to the intervalley momentumforbidden and intravalley spin-forbidden dark trions [22][23][24] . The peak T 1 was recently ascribed to the optical recombination of D − mediated by electron-electron intervalley scattering 22,27 . Several peaks are observed in the lower energy side of the spectrum and are attributed to the replicas of D − and D 0 due to the interaction with chiral phonons. We identify D À i as replicas of D − generated by the interaction with valley phonons i = K 1 ,K 2 ,K 3 ,Γ 5 23,25,28,29 . Theoretical calculations of the energy of these phonon modes, the rationale for the assignment of these peaks, and the discussion on additional peaks observed in the experimental spectrum are reported in the Supplementary Information. The band composition of the dark exciton complexes relevant for the rest of the discussion is illustrated in Fig. 1c-f. Note that the dynamics of D À i and T 1 can be ascribed to the one of D − and D 0 , but, differently from D − and D 0 , their emission is out-of-plane 23 and they appear more clearly in the emission spectrum of Fig. 1b. In the text, we refer to D 0 and D − as dark excitons and dark trions if not otherwise specified.

Diffusion of exciton complexes in WS 2
First, we study how different exciton populations diffuse by measuring the size of their cloud, obtained by spectrally filtering the corresponding peaks as discussed in the Supplementary Information. Figure 2a shows the map of the PL emission from the WS 2 monolayer used for this experiment taken by scanning the sample with galvanometer mirrors in confocal mode (see Fig. S1). During the sample fabrication process, we introduce tensile strain gradients in a large region of the monolayer to create an energy landscape for excitons. The details of the fabrication process and the characterization of strain are discussed later in the text and in the SI. The position in the sample and the direction of the energy gradient are indicated in Fig. 2a. Figure 2b-d is the spatially resolved emission of X 0 , T 1 and D À P = D À K3 + D À Γ 5 , respectively, taken with an EMCCD. Figure 2b shows that the extension of the bright exciton cloud vanishes completely around 4 μm away from the excitation location. The diffusion of bright excitons is driven by their non-uniform concentration at the excitation spot and it is constrained by the short lifetime, in the order of 1-5 ps, and weak interaction 30,31 . Differently from the bright exciton, the clouds of dark species extend over a larger area, and the emission is still detected more than 8 μm away from the excitation location, double the distance of its bright counterpart. A comparison of the intensity profiles is illustrated in Fig. 2e. Two main factors contribute to the longer diffusion of dark excitons: the long lifetime allows them to travel further distances, and the larger interaction energy generates a drift potential that pushes dark excitons away from the excitation location. These features are well suited for the requirements of long transport in quantum technologies.

Density-dependent propagation of dark excitons in WS 2
The strong interaction of dark excitons is further demonstrated by power-dependent measurements. The evolution of the emission spectrum for increasing pump power, reported in Fig. S7 in the SI, shows a smaller redshift due to band renormalization for dark exciton species when compared to the bright counterparts [32][33][34] . This suggests that dark excitons are affected by stronger repulsive interactions. The effect of this interaction is manifested more clearly by the expansion of  19 . Differently from dark excitons, the diffusion length of X 0 is not density-dependent and stays constant at around 1 μm for all pump powers. A similar value for the diffusion length of bright excitons has been already reported 19 and attributed to the combined contribution of a rapid and slower decay rate 36 . However, we note that, while our imaging method is capable of detecting very weak signals, point sources in the sample can appear diffuse, affecting the spatial resolution. A comparison with confocal imaging is discussed in Section 5 and Figure S11 of Supplementary Information.

Exciton diffusion in different energy landscapes
The differential between the diffusibility of dark excitons and other excitonic complexes allows us to analyze how exciton diffusion interacts with a varying potential energy landscape. We note that, despite the large inhomogeneity of the sample (Fig. 2a and Fig. S12), long diffusion occurs in the dark exciton species, indicating robustness against energy barriers due to strain and impurities. To further prove that dark excitons are a resilient way to transport energy and information across samples, we study exciton diffusion in a linear energy gradient. Tensile strain is carefully produced in the encapsulated WS 2 monolayer during the transfer process (more details in the SI) such that a linear energy gradient is created in a region extending over 6 μm. Tensile strain reduces the energy band-gap (E BG ) at the K valley and transforms the energy landscape for excitons whose energy reads E X ðxÞ = E BG ðxÞ À E B . Here we assume that the exciton binding energy (E B ) does not depend on the spatial coordinate x, because strain only weakly affects it in the strain range investigated in this work 37 . Uniaxial strain variations in TMDs have been shown to modify the emission energy of the neutral excitons X 0 with a uniaxial strain gauge factor of −45 meV/% 38,39 . Although this value has been measured at room temperature in previous experiments, it is a good approximation to estimate the strain magnitude in our samples. Figure 4a shows the position-dependent PL spectra measured by exciting and collecting the emission spectra in the same position along one direction. Note that, for the sake of visibility, in all color maps of Fig. 4 the spectra (plotted as columns of the map) are normalized to their maximum, and the energy of bright (dark) exciton complexes is highlighted by red (other colors) circular markers. The variation of intensity of the peaks as a function of space can be appreciated in the raw spectra reported in Supplementary Fig. S13. Figure 4a shows that as the position moves in the direction of increasing applied strain, a clear linear energy shift up to 20 meV is visible for all excitons, confirming the presence of a onedimensional linear strain gradient in the sample from 0 to 0.4%. This strain distribution allows us to study exciton diffusion in both downhill and uphill energy landscapes. We perform energy-resolved diffusion experiments by decoupling detection and excitation so that excitons are excited in one location and their emission spectrum is measured far away from it (see Supplemental Fig. S1). The schematic of the experiment is illustrated in the top panel of Fig. 4b, c. When excitons diffuse away from the excitation spot, they relax toward their minimum energy state before recombining radiatively 28 and their emission energy depends on the local conditions, including the variation of E BG due to strain (Fig. 1a). This process is illustrated in Fig. 5c.
First, we study downhill diffusion by creating the exciton gas in a point of the sample with zero strain corresponding to high potential energy and by measuring the emission spectra at different positions along the strain gradient. Differently from other exciton species, the energy of the negatively charged biexciton (XX À ) and the bright trions (X À T and X À S ) does not change over the entire strain landscape. Due to their low binding energy ( ∼ 20-40 meV), very short lifetime ( ∼ 30 ps), and large mass 24,40 , the diffusion of these species is limited. However, at the excitation location, they have the strongest emission (see Fig. S7), which can be picked by our large NA objective lens even at a distance of a few micrometers. This conclusion is supported by the constant emission energy, showing that the recombination occurs only in the same location of the excitation. However, we observe significant diffusion of bright excitons up to 2.5 μm, in agreement with the diffusion measurements of Fig. 2c. Despite its short lifetime, X 0 has large binding energy, and the tensile strain generates a funneling effect similar to what was observed in previous works 41 . Very good correlation emerges between the energy landscape created by the strain gradient and the emission energy of dark exciton complexes, including I 0 , D 0 , D − , T 1 , D À P . The downhill transport of dark excitons is due to the combination of repulsion interaction and the funneling force F f un = À ∇EðxÞ generated by the strain gradient.
Finally, we study uphill diffusion by exciting the sample in a location with high strain, corresponding to low potential energy. Figure 4c shows that the emission energy of all bright excitons detected away from the excitation spot is constant, despite the increase of potential energy. This indicates that they are not propagating, and they are observed in the spectrum only because the large NA objective lens picks up their strong out-of-plane emission at the excitation spot. This result also confirms the weak interaction energy among bright exciton species in TMDs 10 . Surprisingly, dark exciton complexes show a remarkable blueshift as a function of the distance, indicating that they overcome the potential gradient. Dark excitons can diffuse away from the excitation spot, relax to the energy landscape-which represents the lowest energy state available to them at that particular positionand then recombine radiatively. This observation together with the increase of the diffusion length as a function of pump power indicates the buildup of interaction energy in the dark exciton population. This effect is particularly evident for I 0 , D 0 , T 1 , D À Γ 5 , and D À K3 . This experiment sets a lower limit for the blueshift induced by repulsive interaction to 20 meV, limited by the energy landscape size. The interaction-driven propagation of dark excitons is confirmed in similar diffusion experiments performed at lower excitation powers (Fig. S16) and different energy landscapes, including shorter energy ramps (Fig. S17) and almost flat energy landscapes (Fig. S18).

Dark exciton interaction and relaxation pathways
We estimate the strength of the exciton-exciton interaction with the theoretical model based on Coulomb interactions between excitons described in refs. 42,43. While the direct contribution to the interaction is generally weak and vanishes for small values of center of mass momentum, the exchange contribution is always repulsive and finite 43,44 , with an approximate strength of U ex ∼ a o 2 E b , where E b is the binding energy and a o is the Bohr radius. Therefore, the main source of interaction is provided by the exchange term which is spin and valley dependent 4 . Bright and dark excitons have comparable binding energy 45,46 and radius 26 , and their exchange interaction strength U ex is expected to be similar. Since the overall interaction energy is given by ΔE = n 0 U ex , the longer propagation and the blueshift of dark excitons observed experimentally is attributed to a higher density. To estimate the order of magnitude of the exciton density, we solve a set of coupled rate equations for the population dynamics of bright, intervalley and dark exciton states as illustrated in Fig. 5a. The model accounts for the radiative lifetime 9 , nonradiative phonon-mediated interband transitions 47 , and exciton-exciton annihilation processes 48 . Upon excitation, the bright exciton X 0 can relax towards the I 0 or D 0 state. The X 0 →I 0 transition is induced by the interaction with chiral phonons K 3 which promote the spin-preserved intervalley scattering of electrons in the conduction bands from K to K' (see Fig. 1f). The X 0 → D 0 transition is induced by the interaction with chiral phonons Γ 5 which promote the spin-flipping intravalley scattering of electrons in the conduction bands. In n-doped samples, the transition I 0 → D 0 is mediated by scattering with free electrons. Figure 5b shows that in a large range of generation rates (G), the density of dark excitons is almost three orders of magnitude larger than the one of the bright and the intervalley excitons, indicating that ΔE is expected to be greatest for dark excitons, explaining the experimental observation. We note that the effect of exciton-exciton annihilation is only significant for values of generation rates much larger than the ones used in the experiments. We use a power density of ∼ 5 Á kW cm À2 that, for an absorption efficiency of ∼ 5% 49 , corresponds to a generation rate of G ∼ 2 Á 10 9 cm À2 ps À1 returning a density of D 0 in the order of 8 Á 10 11 cm À2 . For a blueshift of ΔE = 20 meV as in the case of Fig. 4b, c, the corresponding interaction strength is U ex ∼ 2:5 Á 10 À11 meV cm 2 , which is one order of magnitude larger compared to the one measured in bright excitons 50 . This deviation can originate from the different properties between dark and bright excitons, or from the limitations of the model used to estimate the dark exciton density discussed in the SI. However, the calculated density is compatible with the ones extracted from the experimental data when the interaction strength U ex is estimated by using the values of the bright exciton radius and binding energy reported in the literature (more details in the SI) 45,51,52 . The estimated range of experimental values for the dark exciton density is 1 Á 10 12 À 3 Á 10 12 cm À2 . The agreement between the calculated and measured values of the dark exciton density suggests that the long propagation of dark excitons is promoted by strong exciton-exciton interactions due to their high density and long lifetime.
We note that bright single and triplet trions can relax toward the dark trion state via an electron scattering process with a characteristic time similar to the one of the neutral excitons. Therefore, the density of D − is expected to be similar to the one of D 047 . However, due to its the smaller binding energy, the interaction energy experienced by D − should be smaller.
It is worth noting that despite the similar lifetime of I 0 and D 0 , our model returns a density for I 0 which is orders of magnitude smaller than the one of D 0 . While D 0 is the lowest exciton state of the system, the dynamics of I 0 is governed by the fast I 0 → D 0 scattering channel that quickly depopulates I 0 in favor of D 0 . Interestingly, in the experiment of Fig. 4c, I 0 recombines at higher energy with respect to the excitation location despite its low density and weak interaction. Even though it is energetically unfavorable, the transition D 0 → I 0 can occur when the kinetic energy of dark excitons is larger than ∼ 20 meV 47 , compatible to the one observed in our experiments. This observation suggests that I 0 generates from the relaxation of the transported D 0 . Further diffusion measurements over varying energy landscapes and with different excitation powers are reported in Supplementary  Fig. S16-S18.

Discussion
Our polarization measurements (Fig. 1b), consistent with previous works, indicate that dark exciton replicas posses spin-valley coherence 7,23,25 . Although spin-valley polarization cannot be measured directly from D 0 and D − due to their out-of-plane optical transition, it can be observed in their phonon replicas that reveal the valley information of the original dark states. Therefore, due to their robust diffusion over several micrometers, dark excitons are capable of transporting spin information across large areas and inhomogeneous landscapes. This information can then be read out in the far-field via out-of-plane emission generated by phonon or electron interaction. The transport and relaxation pathways for the dark exciton D 0 are schematically illustrated in Fig. 5c. The system is initially excited to create dark excitons as shown in Fig. 5a. The high density of dark excitons favors the strong exciton-exciton repulsive interaction that increases the overall energy of the dark exciton population at the excitation position. This repulsive interaction at the excitation position provides initial momentum to excitons which diffuse away even towards locations of the sample with higher energy landscape. The transported hot dark excitons can then thermalize toward the bottom of their energy dispersion via scattering with low energy Γ acoustic phonons 28 . From this state they can radiatively recombine in the form of dark excitons D 0 , dark trions D − , phonon replica, T 1 or I 0 .
In summary, we have reported the observation of a robust driftdiffusion process in dark excitons in WS 2 that extends over several micrometers in an inhomogeneous energy landscape. We provide evidence that the diffusion is driven by repulsion, stemming from the high exciton density. Due to the introduction of an advanced fabrication process and imaging setup, we are able to study diffusion of dark excitons in a linear potential energy gradient. With these experiments, we show that dark excitons are promising transport means for spin-valley information for applications in quantum information technology.

Sample preparation
2D materials are mechanically exfoliated using a standard scotch tape method on a Polydimethylsiloxane (PMDS) stamp (X4 WF Film from Gel-Pak) and then transferred onto a Si/SiO 2 substrate. We use hBN flakes of thickness around ∼ 20 nm as our bottom and top protective layers. Monolayers of WS 2 are initially exfoliated from the bulk crystals (from HQ Graphene) and then transferred onto the hBN substrate. During this process, strain is imprinted on the monolayer by using a large contact angle between the WS 2 and the hBN layer. Finally, the 2D material stack is encapsulated with the top hBN layer and the resulting heterostructure is baked at 200°for 5 min in air to remove nanobubbles created at the WS 2 /hBN interfaces during the transfer process. More details in the SI.

PL and hyperspectral measurement
PL and spectroscopy measurements are carried out in a home-built confocal microscope setup coupled to a closed-cycle cryostat. PL experiments are performed by exciting the samples non-resonantly with a continuous-wave green laser (532 nm). The laser spot size on the sample is 2 μm. PL maps are taken with a galvanometer mirror scanner in a 4f configuration. The diffraction-limited spatial resolution is approximately 350 nm. An objective lens with NA = 0.9 and free-spacecoupled avalanche photodiodes are used for high-efficiency collection. In detection, the excitation laser is filtered out with a 550-nm long-pass filter. Spectra are obtained by directing the signal to a spectrometer with gratings of 600 G/mm. Images of the exciton clouds are taken with an EMCCD camera in the low noise mode. The complete scheme of the optical setup is in Fig. S1.

Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding authors on request.

Code availability
The codes used to generate the data are available from the corresponding authors upon request.
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